Nuprl Lemma : bag-remove1-member
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)]. (bag-remove1(eq;{x} + bs;x) = (inl bs) ∈ (bag(T)?))
Proof
Definitions occuring in Statement :
bag-remove1: bag-remove1(eq;bs;a),
bag-append: as + bs,
single-bag: {x},
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
unit: Unit,
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
exists: ∃x:A. B[x],
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
label: ...$L... t,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_or: a ↓∨ b,
squash: ↓T
Lemmas referenced :
bag-remove1-property,
bag-append_wf,
single-bag_wf,
bag_wf,
unit_wf2,
list_wf,
permutation_wf,
istype-universe,
deq_wf,
bag-append-cancel,
list-subtype-bag,
subtype_rel_self,
bag-member-append,
bag-member-single,
bag-member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
unionElimination,
pointwiseFunctionalityForEquality,
unionEquality,
sqequalRule,
pertypeElimination,
productElimination,
productIsType,
equalityIsType4,
universeIsType,
because_Cache,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
universeEquality,
inlEquality_alt,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
inlFormation_alt,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. (bag-remove1(eq;\{x\} + bs;x) = (inl bs))
Date html generated:
2019_10_16-AM-11_30_52
Last ObjectModification:
2018_10_11-AM-10_03_28
Theory : bags_2
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